S2T2

Jim Randell 5:03 pm on 9 May 2019 Permalink Reply
Tags: by: Nick MacKinnon ( 52 )   

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  Jim Randell 5:28 pm on 9 May 2019 Permalink | Reply

  Once you’ve worked out the shapes involved, you find that Eve’s answer is 8x^4, and a simple program (or a simple guess) lets us find the appropriate answer.

  Run: [ @repl.it ]

  from enigma import (irange, inf, is_duplicate, is_power, printf)
   
  # consider the length of the side
  for x in irange(1, inf):
    E = 8 * x**4
    if E > 9876543210: break
    if is_duplicate(E): continue
    if not is_power(E, 5): continue
    printf("E={E} [x={x}]")
  

  Solution: Eve’s answer was 32768.

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    Jim Randell 9:08 am on 12 May 2019 Permalink | Reply

    Here’s the analysis to work out the formula for Eve’s answer:

    If we consider a hexagon (with sides of unit length) that has a right angle at a given vertex, then we can’t have another right angle at an adjacent vertex, as it becomes impossible to construct a convex hexagon if we do.

    So, we need only consider a pair of right angles separated by 1 or 2 intermediate vertices.

    Taking the case with 2 intermediate vertices we get a hexagon that looks like this:

    

    The length of the diagonal being (1 + √2).

    In the case with only 1 intermediate vertex we get a hexagon that looks like this:

    

    If the angle XYZ is θ, then the area of the triangle XYZ is given by:

    area(XYZ) = ((√2)/2)sin(θ)

    which is at a maximum when sin(θ) = 1, i.e. θ = 90°.

    And the length of the diagonal d = √3.

    So, for hexagons with side x, the two diagonals have lengths:

    (1 + √2)x
    (√3)x

    Adam’s value (the sum multiplied by the difference) is:

    A = (1 + √2 + √3)x(1 + √2 − √3)x
    A = 2(√2)x²

    And Eve’s value is the square of this:

    E = 8x⁴

    And we are told E is an exact power of 5.

    Choosing x = 8 gives E = 8⁵ = 32768, which has five different digits as required.

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      Jim Randell 10:46 pm on 12 May 2019 Permalink | Reply

      For completeness:

      The general case of the first hexagon (with right angles separated by two vertices) is a shape like this:

      

      The red line bisects the hexagon into two identical shapes (by rotation), but in general is not a line of reflective symmetry.

      Again the area of the triangle XYZ is given by the formula:

      area(XYZ) = ((√2)/2)sin(θ)

      and so the maximum area is achieved when sin(θ) = 1, i.e. θ = 90°.

      Which gives us the diagram originally presented (where the red line is a line of reflective symmetry).

      So both gardens have the same area, being composed of two (1, 1, √2) triangles and two (1, √2, √3) triangles. Giving a total area of (1 + √2).

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